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Update math notation for initial condition gradients in README.md
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yoyolicoris committed Apr 15, 2024
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### Gradients for the initial condition $`y_t|_{t \leq 0}`$

The initial conditions provide an entry point at $t=0$ for filtering, as we cannot evaluate $t=-\infty$.
Let us assume $A_{t, :}|_{t \leq 0} = 0$ so $y_t|_{t \leq 0} = x_t|_{t \leq 0}$, which also means $\frac{\partial \mathcal{L}}{y_t}|_{t \leq 0} = \frac{\partial \mathcal{L}}{x_t}|_{t \leq 0}$.
Let us assume $`A_{t, :}|_{t \leq 0} = 0`$ so $`y_t|_{t \leq 0} = x_t|_{t \leq 0}`$, which also means $`\frac{\partial \mathcal{L}}{y_t}|_{t \leq 0} = \frac{\partial \mathcal{L}}{x_t}|_{t \leq 0}`$.
Thus, the initial condition gradients are

$$
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$$

In practice, we pad $N$ and $N \times N$ zeros to the beginning of $\frac{\partial \mathcal{L}}{\partial \bf y}$ and $\mathbf{A}$ before evaluating $\frac{\partial \mathcal{L}}{\partial \bf x}$.
The first $M$ outputs are the gradients to $y_t|_{t \leq 0}$ and the rest are to $x_t|_{t > 0}$.
The first $M$ outputs are the gradients to $`y_t|_{t \leq 0}`$ and the rest are to $`x_t|_{t > 0}`$.

### Time-invariant filtering

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